Volterra – Fredholm integral equations (VFIEs) have a massive interest from researchers recently. The current study suggests a collocation method for the mixed Volterra - Fredholm integral equations (MVFIEs)."A point interpolation collocation method is considered by combining the radial and polynomial basis functions using collocation points". The main purpose of the radial and polynomial basis functions is to overcome the singularity that could associate with the collocation methods. The obtained interpolation function passes through all Scattered Point in a domain and therefore, the Delta function property is the shape of the functions. The exact solution of selective solutions was compared with the results obtained from the numerical experiments in order to investigate the accuracy and the efficiency of scheme.

Recently, there has been an increasing interest in the study of singular and perturbed systems. In this paper design fast feed forward neural network to present a method to solve two dimensions singularly perturbed integrodifferential and integral equations. Using a multi-layer having one hidden layer with 7 hidden units (neurons) and one linear output unit the sigmoid activation of each unit is radial basis function and Levenberg -Marquardt (trainlm) training algorithm. Finally the results of numerical experiments are compared with the exact solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme..

Recently, there has been an increasing interest in the study of singular and perturbed systems. In this paper we propose a point interpolation meshless method for solving two-dimensional singularly perturbed integro-differential equations. The results of numerical experiments show that the numerical scheme is very effective and convenient for solving a large number of singularly perturbed problems with high accuracy.

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The aim of this paper  is present a new numerical method for solvingThree Dimensions Volterra Integral Equations using artificial neural network by design multilayer feed forward Neural Network. A multi- layers design in our proposed method consist of a hidden layer having seven hidden units. and one linear output unit. Linear Transfer function used as each unit and using Levenberg- Marquardtalgorithmtraining. Moreover, examples on three- dimensional Volterra integral equations carried out to illustrate the accuracy and the efficiency of the presented method. In addition, some comparisons among proposed method and Shifted Chebyshev Polynomials method and Reduced Differential Transform Method are presented.   http://dx.doi.org/10.25130/tjps.23.2018.176